Concept

Rational points

The points of a curve whose coordinates are both fractions. A circle through one such point has infinitely many, found by lines of rational slope, and whether a curve has any at all is a question about primes and remainders.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

number · Pythagoras

From a rational point on x² + y² = 17 to a whole one. The circle of radius √17 on the integer lattice, a rational point on it, and 4 reflections through nearest lattice points, the denominators 3757, 205, 25, 5, 1, ending at (1, 4).

A fraction on the circle forces a whole point

If a number is a sum of two squares of fractions, it is a sum of two squares of whole numbers. Draw the circle, mark the rational point, join it to the nearest lattice point and follow the line to where it meets the circle again: the new point is rational too, with a smaller denominator. Repeat, and the denominators fall until they reach one. The argument needs no primes at all — only the fact that every point of the plane is within distance one of the lattice.

number · Sums of two squares

Named alongside it

The objects these essays reach for when they reach for this one.

Descent Pythagorean triples Bijection Conic Hypotenuse Lattice Norm Parametrisation Primitive triple Projection Quadratic form Right triangle

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